PlanetPhysics/Center of Abelian Category

Let ${\mathcal {A}}$ be an abelian category. Then one also has the identity morphism (or identity functor) $id_{\mathcal {A}}:{\mathcal {A}}\to {\mathcal {A}}$. One defines the center of the Abelian category Failed to parse (syntax error): {\displaystyle \mathcal{A } } by $Z({\mathcal {A}})=End(id_{\mathcal {A}}).$

One can show that the center is $Z(CohX)\cong {\mathcal {O}}((X)$ for any algebraic variety where ${\mathcal {O}}(X)$ is the ring of global regular functions on $X$ and ${\mathbf {C} oh}(X)$ is the Abelian category of coherent sheaves over $X$.

One can show also prove the following lemma. \begin{theorem} {\mathbf Associative Algebra Lemma}

If $A$ is a associative algebra then its center $Z(A-mod)=ZA.$ \end{theorem}